Proof Of Lemma Research Articles

Erratum to “Navier–Stokes equations in a curved thin domain, Part III: thin-film limit”

There are two flaws in the third part [3] of our study on the Navier--Stokes equations in a curved thin domain. The first one is that the form of Ladyzhenskaya's inequality on a closed surface taken from the arXiv version [2] of the second part of our study is incorrect. The second one is that a wrong claim is used in the proofs of lemmas in [3, Section 6.3] which give estimates for the weighed potential part of a tangential surface vector field. This erratum corrects these flaws.

Виявлення прихованих перiодичностейв моделях з дискретним часом та сильно залежним випадковим шумом

Trigonometric regression models take a special place among various models of nonlinear regression analysis and signal processing theory. The problem of estimating the parameters of such models is called the problem of detecting hidden periodicities, and it has many applications in natural and technical sciences. The paper is devoted to the study of the problem of detecting hidden periodicities in the case when we observe only one harmonic oscillation with discrete time, where random noise is a local functional of Gaussian random sequence with singular spectrum. In particular, the random sequence in the model can be strongly dependent. For estimation of unknown parameters the periodogram estimator is chosen. Sufficient conditions of the consistency of the amplitude and angular frequency periodogram estimator of the model described above are obtained in the paper. The proof of Lemmas 1 and 2 gave an important asymptotic properties of the random noise functional related to the periodogram estimator which necessary for the proof of the main results. Series expansion of random noise in terms of Hermite polynomials and the Diagram formula are main tools that were used to obtain this lemmas.

Correction to: Conservative and Semiconservative Random Walks: Recurrence and Transience

The aim of this note is to correct the errors in the formulation and proof of Lemma 4.1 in [1] and some claims that are based on that lemma.

Corrigendum: On the complexity of finding first-order critical points in constrained nonlinear optimization

In a recent paper (Cartis et al. in Math Prog A 144(2):93---106, 2014), the evaluation complexity of an algorithm to find an approximate first-order critical point for the general smooth constrained optimization problem was examined. Unfortunately, the proof of Lemma 3.5 in that paper uses a result from an earlier paper in an incorrect way, and indeed the result of the lemma is false. The purpose of this corrigendum is to provide a modification of the previous analysis that allows us to restore the complexity bound for a different, scaled measure of first-order criticality.

Corrigendum: Further refinements of the GL(2) converse theorem

We correct an error in the proof of Lemma 2.4. The analysis of possible poles applies unchanged and shows that if L ( s , π ∞ ⊗ ω ∞ ) and L ( 1 − s , π ˜ ∞ ⊗ ω ∞ − 1 ) have a common pole, then x ( ω ) lies in a finite union of hyperplanes in H. Applying the affine transformation x ↦ M x + b ( ω ) and noting that b ( ω ) lies in the Z-linear span of finitely many vectors, we see that y ( ω ) lies in a finite union of sets of the form ( V + A ) ∩ Q n , where V ⊂ R n is a proper subspace and A ⊂ R n is a finitely generated subgroup. Applying the lemma below, we conclude that there is a non-negative integer m and non-zero vectors w 1 , ... , w m ∈ Z n such that w j · y ( ω ) ∈ Z holds for at least one j ⩽ m whenever L ( s , π ∞ ⊗ ω ∞ ) and L ( 1 − s , π ˜ ∞ ⊗ ω ∞ − 1 ) have a common pole, and the rest of the proof goes through as before. Lemma.Let r , s , n ∈ Z ⩾ 0 with n > r , and consider vectors u 1 , ... , u r , v 1 , ... , v s ∈ R n . Then there is a non-zero vector w ∈ Z n such that Proof.Let x = ( x 1 , ... , x n ) be a typical element of ( R u 1 + ⋯ + R u r + Z v 1 + ⋯ + Z v s ) ∩ Q n and choose a non-zero vector z = ( z 1 , ... , z n ) ∈ R n orthogonal to u 1 , ... , u r . Then Next, consider the group

A correspondence between ideals and z-filters for certain rings of continuous functions – some remarks

Let X be a completely regular Hausdorff topological space and A(X) a ring lying between C⁎(X) and C(X). A correspondence ZA between ideals of A(X) and the z-filters on X was initiated by Redlin and Watson in 1987 and was further investigated by Byun and Watson in a paper published in Topology and its Applications in 1991. In the last mentioned paper, the authors have established a lemma which reads that for any two rings A(X) and B(X) lying between C⁎(X) and C(X) with B(X)⊆A(X) and for any ideal I of A(X), ZA[I]=ZB[I∩B(X)]. We point out an error in the proof of this lemma. The authors have used this lemma to prove a theorem, which says that (a) if M is a maximal ideal of A(X) then ZA[M] is contained in a unique z-ultrafilter on X and (b) if F is a z-ultrafilter on X, then ZA−1[F] is a maximal ideal of A(X). The authors have given a correct proof of part (b) of this result, in a more general context, in a later article [Redlin and Watson, 1997]. We give a correct proof of the above lemma and generalize part (a) of the above theorem to prime ideals. Lastly we show that if A(X)≠C(X), then there exists a non-maximal prime ideal in A(X).

Correction to a proof in the article “Patching and admissibility over two-dimensional complete local domains”

The proof of Lemma 1.8 of the article in the title is incorrect. We supply an alternate argument for Proposition 1.10, whose proof invoked that lemma.

Corrigendum to the paper ‘Linear Time Varying Model Predictive Control and its Application to Active Steering Systems: Stability Analysis and Experimental Validation’ (International Journal of Robust and Nonlinear Control 2008; 18 (8):862-875)

This note is a corrigendum of the paper ‘Linear Time Varying Model Predictive Control and its Application to Active Steering Systems: Stability Analysis and Experimental Validation’, published on the volume 18, issue 8, pages 862–875 of the International Journal of Robust and Nonlinear Control in 2008. Next we point out a technical error in the proof of Lemma 2 of the paper, and provide the corrected version of the lemma. Copyright © 2011 John Wiley & Sons, Ltd.

A correction to “A non-implication between fragments of Martin’s Axiom related to a property which comes from Aronszajn trees”

In the paper A non-implication between fragments of Martin’s Axiom related to a property which comes from Aronszajn trees (Yorioka, 2010 [1]), Proposition 2.7 is not true. To avoid this error and correct Proposition 2.7, the definition of the property R 1 , ℵ 1 is changed. In Yorioka (2010) [1], all proofs of lemmas and theorems but Lemma 6.9 are valid about this definition without changing the proofs. We give a new statement and a new proof of Lemma 6.9.

A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids

You have accessMoreSectionsView PDF ToolsAdd to favoritesDownload CitationsTrack Citations ShareShare onFacebookTwitterLinked InRedditEmail Cite this article Mihăilescu Mihai and Rădulescu Vicenţiu 2011A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluidsProc. R. Soc. A.4673033–3034http://doi.org/10.1098/rspa.2011.0070SectionYou have accessCorrectionsA multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids Mihai Mihăilescu Mihai Mihăilescu Google Scholar Find this author on PubMed Search for more papers by this author and Vicenţiu Rădulescu Vicenţiu Rădulescu Google Scholar Find this author on PubMed Search for more papers by this author Mihai Mihăilescu Mihai Mihăilescu Google Scholar Find this author on PubMed Search for more papers by this author and Vicenţiu Rădulescu Vicenţiu Rădulescu Google Scholar Find this author on PubMed Search for more papers by this author Published:23 February 2011https://doi.org/10.1098/rspa.2011.0070This article corrects the followingResearch ArticleA multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluidshttps://doi.org/10.1098/rspa.2005.1633 Mihai Mihăilescu and Vicenţiu Rădulescu volume 462issue 2073journalTitle Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences31 March 2006The goal of this correction is to correct a mistake that appears in the proof of both lemma 3.9 and 3.10 in the article Mihăilescu & Rădulescu (2006).The mistake in the proof of lemma 3.9 and 3.10 is mainly owing to the fact that we wrongly computed the expression of function G(x,t) when t>u1(x). The correct computation reads as follows: 1.1 We remember the result of Mihăilescu & Rădulescu (2006, lemma 3.9) and provide the correct proof.Lemma 3.9There exists ρ∈(0,∥u1∥) and a > 0 such that J(u) ≥ a, for all u∈E with ∥u∥=ρ.Proof.Let u∈E be fixed, such that ∥u∥<1. It is clear that Define If x∈Ω\Ωu then and we have If x∈Ωu∩{x;u1(x)<u(x)<1} then Define Thus, provided that ∥u∥<1 by condition (A5), the above estimates and relation (1.5) we get 1.2 Since , it follows that p+<p☆(x) for all x∈Ω. Then, there exists q∈(p+,Np−/(N−p−)) such that E is continuously embedded in Lq(Ω). Thus, there exists a positive constant C>0 such that Using the definition of G and the above estimate, we obtain 1.3 where D and D1 are two positive constants. Combining inequalities (1.2) and (1.3), we find that for a small enough we have and taking into account that q>p+, we infer that the conclusion of this lemma holds true. ■Regarding the proof of Mihăilescu & Rădulescu (2006, lemma 3.10) it can be carried out with the same arguments as in the original proof, but the expression of G(x,t) with t>u1(x) should be replaced in the proof with the one given by relation (1.1).FootnotesThis journal is © 2011 The Royal SocietyReferencesMihǎilescu M.& Rǎdulescu V.. 2006A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids. Proc. R. Soc. A 462, 2625-2641doi:10.1098/rspa.2005.1633 (doi:10.1098/rspa.2005.1633). Link, Google Scholar Previous ArticleNext Article FiguresRelatedReferencesDetailsRelated articlesA multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids31 March 2006Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences This Issue08 October 2011Volume 467Issue 2134 Article InformationDOI:https://doi.org/10.1098/rspa.2011.0070Published by:Royal SocietyPrint ISSN:1364-5021Online ISSN:1471-2946History: Published online23/02/2011Published in print08/10/2011 License:This journal is © 2011 The Royal Society Citations and impact PDF Download Subjectsapplied mathematicsdifferential equations

A sum theorem for (FPV) operators and normal cones

In his 2008 book “From Hahn–Banach to Monotonicity”, S. Simons mentions that the proof of Lemma 41.3, which was presented in the previous edition of his book “Minimax and Monotonicity” in 1998, is incorrect, and one does not know if this lemma and its consequences are true. The aim of this short note is not only to give a proof to the mentioned lemma but also to improve upon it by relaxing one of its assumptions.

Corrigendum to ``West's problem on equivariant hyperspaces and Banach-Mazur compacta''

In our article [1], on p. 3389, the definition of the weak topology of the G-nerve N(JA) contains a gap. Namely, it is claimed there that the topology on N(JA) induced from J is the weak one, which is false. The author apologizes for this mistake. Nevertheless, in the proofs of Lemmas 4.2, 4.4 and 5.2, where the topology of M (JA) is essential, in fact the right weak topology of M (JA) was applied. Thus, all of the proofs given in [1] are correct and complete. Using the notation and references adopted in [1], the above-mentioned gap in the definition of the topology of the G-nerve M (JA) may be filled by replacing the text on p. 3389 starting in line 26 and ending in line 35, by the following. For every simplex L = (/?o, , Un) C M (JA), set

Classically-controlled Quantum Computation

It is reasonable to assume that quantum computations take place under the control of the classical world. For modelling this standard situation, we introduce a Classically-controlled Quantum Turing Machine (CQTM) which is a Turing machine with a quantum tape for acting on quantum data, and a classical transition function for a formalized classical control. In CQTM, unitary transformations and quantum measurements are allowed. We show that any classical Turing machine is simulated by a CQTM without loss of efficiency. Furthermore, we show that any k-tape CQTM is simulated by a 2-tape CQTM with a quadratic loss of efficiency. The gap between classical and quantum computations which was already pointed out in the framework of measurement-based quantum computation (see [S. Perdrix, Ph. Jorrand, Measurement-Based Quantum Turing Machines and their Universality, arXiv, quant-ph/0404146, 2004]) is confirmed in the general case of classically-controlled quantum computation. In order to appreciate the similarity between programming classical Turing machines and programming CQTM, some examples of CQTM will be given in the full version of the paper. Proofs of lemmas and theorems are omitted in this extended abstract.

A sharp form of Whitney’s extension theorem

3. Order relations involving multi-indices 4. Statement of two main lemmas 5. Plan of the proof 6. Starting the main induction 7. Nonmonotonic sets 8. A consequence of the main inductive assumption 9. Setup for the main induction 10. Applying Helly's theorem on convex sets 11. A Calder6n-Zygmund decomposition 12. Controlling auxiliary polynomials I 13. Controlling auxiliary polynomials II 14. Controlling the main polynomials 15. Proof of Lemmas 9.1 and 5.2 16. A rescaling lemma 17. Proof of Lemma 5.3 18. Proofs of the theorems 19. A bound for k# References

Erratum: Stabilization of solutions to nonlinear Schrödinger equations

The proof of lemma 5.2 in [1] contains several mistakes. Nevertheless, thestatement is correct and is proven in an elementary fashion, correctly this time, in[3, lemma 2.4], which is in this issue of the journal.In the proof of corollary 3.2 in [1], we misquoted from Kato’s textbook on per-turbation theory. In fact, it is not true that, as asserted in [1], corollary 3.2 followsfrom lemma 3.1. However, corollary 3.2 of [1] is a consequence of proposition 3.1in [2] or, under more general hypotheses that allow eigenvalues embedded in thecontinuous spectrum and for space dimension 3, proposition 4.1 in [3].

A correction to the proof of a lemma in "The capacity of wireless networks"

For original paper see "The capacity of wireless networks", Gupta and Kumar, ibid., vol. 46, p. 388-404 (2000). The proof of lemma 4.8 is corrected.

TOWARD A CLASSIFICATION OF ALMOST HOMOGENEOUS MANIFOLDS III — SL(2)×C* ACTIONS

In this paper we finished the classification of SL(2, C)×Gm almost homogeneous projective 3-folds which we started in [8, 9]. For some technical reasons we leave the proofs of Propositions 1.2 and 1.3 to [8] and that of Propositon 2.7 to [9]. But we include all other proofs, especially those of Propositions 2.1 and 2.2 in this paper and try to give an intuitive picture of the process. The proofs of Lemmas 2.1 and 2.2 as well as those of Sec. 3 (Theorems 3.1–3.4) are the main results of this paper. Most of this work was done in the fall of 1991. I thank those who had tried their best to understand my proofs, for their helpful comments and criticisms.

Comments on "A weighted mixed-sensitivity H/sub /spl infin//-control design for irrational transfer matrices"

This paper shows briefly that, in the above paper by Curtain et al. (see ibid. vol.41 (1996)), there is a slight mistake in the proof of lemma 11.3 and in formula (17) of lemma 111.3. Corrections for formula (17) and a lemma related to it are given.

A self-avoiding walk model of random copolymer adsorption

There is an error in the proof of lemma 3.4 which can be remedied by using (3.22) and (3.13) directly. The proof of lemma 3.2 can also be simplified. Please see PDF for details.

An Axiomatization of the Core of Market Games: A Correction

The proof of Lemma 4.7 in Peleg (Peleg, B. 1989. An axiomatization of the core of market games. Math. Oper. Res. 14 448–456.) is incorrect. However, if we add the following innocent axiom, the proof of the lemma becomes straightforward.